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Abstract

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals - these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Methods

Sixty-four 14-yr-old children completed this task twice at an interval of approximately 60 min, as the first and last sub-tasks in a larger test battery from the longitudinal study. Each run of the task lasted 5 min. Subjects viewed dot arrays on a computer screen and judged whether there were more blue or more yellow dots (download example screenshots). For each trial, pressing the space bar initiated a 250 ms blank-screen delay followed by a 200 ms appearance of an array of intermixed blue and yellow dots. After the array had disappeared, subjects had an unlimited amount of time to indicate their response by pressing a colour-coded keyboard button and saying the name of the more numerous colour aloud. Click play to view some trials. The correct answer is given after each trial, though subjects never saw these answers.

Results

Figure 1 | Method and group performance. a, A representation of the trial from the numerical discrimination task. b, Group performance and modelled best-fit for all trials in the numerical discrimination task. c, Histogram of w, the acuity of the ANS, for the sample (n=64), as determined by the psychophysical model for each subject.

Figure 2 | Regressions. a, b, Linear regression of the standard score for each subject on the TEMA-2 test (a) or on the WJ-Rcalc test (b) of symbolic maths achievement and the acuity of the ANS (w). Each diamond represents a single subject. For TEMA-2 and WJ-Rcalc, higher numbers indicate better performance, whereas for the Weber fraction, lower numbers indicate better performance.

Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the Number Sense: The Approximate Number System in 3-, 4-, 5-, and 6-Year-Olds and Adults. Developmental Psychology, 44 (5). [Demo]

Halberda, J., Taing, L. & Lidz, J. (2008). The development of “most” comprehension and its potential dependence on counting-ability in preschoolers. Language Learning and Development. [Demo]